# Minimum non-zero singular value of $ABC$ where $A$ and $C$ are diagonal and $B$ is semi-unitary

Consider two diagonal matrices $\mathbf{A} = \mathrm{diag}(a_1,\ldots,a_k) \in \mathbb{R}^{k\times k}$ and $\mathbf{C} = \mathrm{diag}(c_1,\ldots,c_k) \in \mathbb{R}^{k\times k}$, and a semi-unitary matrix $\mathbf{B} \in \mathbb{R}^{k\times k}$, $\mathbf{B}\mathbf{B}^T = \mathbf{I}_k$. Is there any bound on the minimum non-zero singular value of $\mathbf{A}\mathbf{B}\mathbf{C}$?